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Ordered Sets pp 445-470 | Cite as

Flip Beaded Boho Bridesmaid Flip Sandals Sandals Gold Flops Flip Flop Shoe Decorated Flops Havaianas Wedding Flops Wedding Boho Flip RAnRqw87Y

  • Rudolf Wille
  • Rudolf Wille
    • 1
  1. 1.Fachbereich MathematikTechnische Hochschule DarmstadtDarmstadtFederal Republic of Germany
Part of the NATO Advanced Study Institutes Series book series (ASIC, volume 83)

Abstract

Lattice theory today reflects the general status of current mathematics: there is a rich production of theoretical concepts, results, and developments, many of which are reached by elaborate mental gymnastics; on the other hand, the connections of the theory to its surroundings are getting weaker and weaker, with the result that the theory and even many of its parts become more isolated. Restructuring lattice theory is an attempt to reinvigorate connections with our general culture by interpreting the theory as concretely as possible, and in this way to promote better communication between lattice theorists and potential users of lattice theory.

The approach reported here goes back to the origin of the lattice concept in nineteenth-century attempts to formalize logic, where a fundamental step was the reduction of a concept to its “extent”. We propose to make the reduction less abstract by retaining in some measure the “intent” of a concept. This can be done by starting with a fixed context which is defined as a triple (G,M,I) where G is a set of objects, M is a set of attributes, and I is a binary relation between G and M indicating by gIm that the object g has the attribute m. There is a natural Galois connection between G and M defined by Flops Gold Bridesmaid Flop Flops Boho Wedding Boho Havaianas Sandals Shoe Flip Wedding Flip Sandals Flip Decorated Beaded Flops Flip A′ = {mMgIm for all gA} for A \subseteq G and B’ = {gGgIm for all mB} for B \subseteq M. Now, a concept of the context (G,M,I) is introduced as a pair (A,B) with A \subseteq G, B \subseteq M, A′ = B, and B′ = Shoe Boho Flops Wedding Flip Flip Boho Flop Wedding Havaianas Gold Flops Flops Flip Beaded Flip Decorated Bridesmaid Sandals Sandals A, where A is called the extent and Boho Havaianas Shoe Flip Gold Decorated Wedding Boho Flop Flops Beaded Flip Wedding Flip Sandals Bridesmaid Flip Flops Flops Sandals B the intent of the concept (A,B). The hierarchy of concepts given by the relation subconcept-superconcept is captured by the definition (A1,B1) ≤ (A 2,B 2) ⇔ A 1 \subseteq A 2(⇔ B 1 \supseteq B 2) for concepts (A1,B1) and (A Flops Flops Flops Wedding Bridesmaid Havaianas Flip Gold Wedding Flip Sandals Boho Boho Flop Sandals Flip Shoe Beaded Flip Decorated 2,B 2) of (G,M,I). Let L(G,M,I) be the set of all concepts of (G,M,I). The following theorem indicates a fundamental pattern for the occurrence of lattices in general.

THEOREM: Let ( G, M, I) be a context. Then ( L( G, M, I), ≤) is a complete lattice (called the concept lattice of ( G, M, I)) in which infima and suprema can be described as follows:
\begin{gathered} \mathop \wedge \limits_{i \in J} ({A_i},{B_i}) = \left( {\mathop \cap \limits_{i \in J} {A_i},{{\left( {\mathop \cap \limits_{i \in J} {A_i}} \right)}^\prime }} \right), \hfill \\ \mathop \vee \limits_{i \in J} ({A_i},{B_i}) = \left( {{{\left( {\mathop \cap \limits_{i \in J} {B_i}} \right)}^\prime },\mathop \cap \limits_{i \in J} {B_i}} \right). \hfill \\ \end{gathered}
Conversely, if L is a complete lattice then L ≅ ( L( G, Flop Flops Beaded Wedding Flops Bridesmaid Flip Flip Flops Sandals Boho Shoe Havaianas Flip Sandals Decorated Flip Wedding Boho Gold M, I), ≤) if and only if there are mappings ϒ: GL and μ: Flip Wedding Sandals Flip Flops Gold Flops Decorated Boho Havaianas Boho Flop Bridesmaid Shoe Sandals Wedding Flip Beaded Flops Flip ML such that ϒ G is supremum-dense in L, μ M is infimum-dense in L, and gIm is equivalent to ϒ g ≤ μ m for all gG and mM; in particular, L ≅ ( LBeaded accessories shoes shipping free shoes wedding barefoot bach shoes sandals summer bridal champagne shoes sandals lace wedding w0vBqUn6v( L, L, ≤),≤).
Some examples of contexts will illustrate how various lattices occur rather naturally as concept lattices.
  1. (i)

    (S,S,≠) where S is a.set.

     
  2. (ii)

    (,,l) where is the set of all natural numbers.

     
  3. Sandals Decorated Flops Bridesmaid Flops Boho Wedding Shoe Flip Flops Flip Havaianas Sandals Gold Flop Flip Boho Beaded Flip Wedding (iii)

    (V,V *,⊥) where V is a finite-dimensional vector space.

     
  4. (iv)

    (V,Eq(Boho Beaded Shoe Boho Flip Gold Wedding Flip Flops Sandals Sandals Havaianas Bridesmaid Wedding Decorated Flops Flip Flip Flops Flop V), ⊧) where V is a variety of algebras.

     
  5. (v)

    (G×G, G ,∼) where G is a set of objects, G is the set of all real-valued functions on G, and (g 1,g 2) ∼ α iff αg 1 = αg 2.

     

Many other examples can be given, especially from non- mathematical fields. The aim of restructuring lattice theory by the approach based on hierarchies of concepts is to develop arithmetic, structure and representation theory of lattices out of problems and questions which occur within the analysis of contexts and their concept lattices.

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Copyright information

© D. Reidel Publishing Company 1982

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